STOP Early Journal Content on JSTOR, Free to Anyone in the World This article is one of nearly 500,000 scholarly works digitized and made freely available to everyone in the world by JSTOR. Known as the Early Journal Content, this set of works include research articles, news, letters, and other writings published in more than 200 of the oldest leading academic journals. The works date from the mid-seventeenth to the early twentieth centuries. We encourage people to read and share the Early Journal Content openly and to tell others that this resource exists. People may post this content online or redistribute in any way for non-commercial purposes. Read more about Early Journal Content at http://about.jstor.org/participate-jstor/individuals/early- journal-content . JSTOR is a digital library of academic journals, books, and primary source objects. JSTOR helps people discover, use, and build upon a wide range of content through a powerful research and teaching platform, and preserves this content for future generations. JSTOR is part of ITHAKA, a not-for-profit organization that also includes Ithaka S+R and Portico. For more information about JSTOR, please contact support@jstor.org. t 417 ] XXVI. A new Method of finding the equal Roots of an Equation, by Dhijion. By the Rev. John Hellins, Curate of Con- ftantine, in Cornwall; communicated by Nevil Maikelyne, D. D. F. R. S. and AJironomer Royal. Read June 20, 1783. ^" , ^HE following theorems are a production of juvenile . years. They were invented about twelve years ago, when algebra was my favourite ftudy ; and one of them (the flrfl:) was published as a fpecimen of this method of extracting 'the equal roots of an equation about ten years ago. Since that time my avocations have left me but very little leifure for im- proving any invention of this kind. Thefe theorems, then, are in their crude ftate ; however, fuch as they are, I flatter myfelf,they will afford an eafier folution of equations that have equal roots than is generally known, and be acceptable to the ingenious algebraift. THEOREM I. If the cubic equation x 3 —px* + qx — r==o has two equal roots s each of them will be (x) zzJoLzSL- DEMONSTRATION. Call the three roots #, a, and b; then, by the compofitioa of equations, we mall have x 3 — 1 ^ x* +™ Ax >~aab 2:04 where Vol, LXXH. I i i %& 4,i 8 Mr. BELLiNs's M&fyod of finding the ?$Jrhzs.pi r aaJp 2ab~.c[, and aab~r ; which values being written m our theorem, we have x ( :=. IXZjLi. i zz v 1pp-~bq ' 2.aaa + 4,aab + aab+ 2ab,b—qaqir laaq — ^aab-^labb @, j-r t-, Saa-i-8af+.itib-~ baa — izab 9.qa~4qb-^-2bi> ~~' ' *^- "' : E.XAME;L:E I. If the equation x T +5^ - 32.Y -f.36 = has two equal roots^ it is propofed to find them by theabove theorem. , H ere p ■==,. — $, q= --32, and, r~ — 36 ; thefe values being written in the theorem, we have ,~^ x ~3 2 ~9 x ~t . -, l6o + 3^ 2X25 r 6x--32 50+49.%,. =-—=?. 2$ which being written for x, the equation becomes 8+20 — 64 + 36, which is evidently =-o; cpnfequently 2 and 2 are roots of it., Otherwife, 2, the value of a; given by the theorem, being written for it in the quadratic equation 3?*+ \ox~..%z = 0, the refult is 1 2 +20 -T..32 ==,©. Or, dividing the given cubic by, the quadratic # - 2I 2 , we, have x* -.4* + 4) x 3 + 5** -, 32,* + 36 (* +.9 ; therefore the three. roots are 2, 2, and —9* E ; X,.AyM BO^J&vIL Given x 3 +J2& - g? = 0, aa equation, which.Ms, equal roots^ to find them. Hereto, and the theorem gives ~: 36poa *4? -.~^p whkh ° r 200x9261 ax' vv *" 1 -* 1 . value being written for x the equation vaniflies. T,;H, ; &0.it.E. M tqual Roots sf an Epiatim iy^lvpm. 4^ T H EOIEM ». If the biquadratic equation x*~px s ^-qx l >~rx + siZQ has two equal roots, make A = ^^ '^B-PSZ^.Ctt^i, andBzz -, and you will have x- D--J 4A+3/.' J " A-C A fynthetical demonftration of this theorem would be very long: the investigation is as follows. It has been demonstrated by the writdrs on algebra, that, if a biquadratic equation, as x*—px % + qx l ^rx-\-s~o, has two equal roots, one of them may be had from the equation 4* 3 — . %px r + zqx - r s o. Multiply this equation by x, and the original one by 4, and take the difference of the two, which will be px* — 2^a? z + $rx — 45- =: o. Again, if this equation be multi- plied by 4, and the other cubic by p, and their difference taken > we mall have 3pp*~ Sq xx* + izr~ zpq x x+pr - i6j , =:o, or x * + ™pW x+ P>^ :zo or *'+A* + Bito, putting A and B for the known quantities in the fecond and third terms. Now multiply this equation by 4.x, and take the firfl cubic from it* and we (hall have 4A + 3J& xx^ + ^B — zq x x + r— o, which being divided by 4A + 3/>, and C and D put equal to $^* nd ^Fy> ref P e & ivel y> gives x* + Cx + t) = o', and this equation being taken from the other quadratic, there remains D-B A-Cx» + B-D-o; confequently x =fe -T— • ^E. I. corollary i. From the above in vefhigation it appears, that one of the equal roots may alfo be obtained from either of- theife two-quadratic equations, of which the firft feems moft eligible, I i i 2 as 420 Mr. itellins's Met bod- of finding the as the co-efficients of it are lefs complex than thofe of the: other ; $pp — 8q x x*+ I2r -~ 2pq x x +pr -~\ts~ Oy. and 4A + 3^ x -x* + 4B — zq x x-hr = o. And thefe* when p~Oy become - 8qx x + 1 %rx — 1 6s — q, and - 5~x ■$<■£-- 2a xx + r =a. Of 5£ ■ * or X — i-#+ — = o> 3* 6 eoROL. 2. If both p and 5- vanifh, then, from either of the quadratics. we get x=s.~, perfectly agreeing with the cubic p>x 3 — zqx* +3rx — 45 — o, which, when^ and q vanifh,. becomes $fx —. 4s =*tO. And this equation, is of ufe ; beeaufe, in this- cafe, the theorem fails,, one of the divifors being — a. corojl. 3.. From the equation 4.x 3 — $px*+ %qx — rzz o, which* when p and % vani(h, becomes 4^ 3 -r=:o, we alfb get *.= ~\f - $, another. expreffion, of the fame value of x. corol. 4. When r = o, D = o, and from the equation **"+,. C* + D =fco, we have x =.- - C. EX A MP L E> I: If the equation x** - q#* + 4*+ 12 = has equal roots, ifeis ^fopofed to find them* Here equal Roots of an Equation by DiviJtctH. 42: Here / « o, q = - 9, r= ■■■ - 4, and s — 1 z ; and A becomes = ll y ~ 4 ~ B D , D-B ~8x- -9 3 :*■ _-i6 x- -8x -9 __8 3 r —5 _ 4X T ! +3 8 _„ 4 ? 4 X — -2 J, _-4 —8 ~~ 3 - 3 z ^ 3 - + "" 1 ^'+32 8 + 33" So _ 25 2 J 4. which being written for x, the equation becomes. 16-36 + 8 + 12 = 0; therefore 2 is one of the roots. The fame value of x may be difcovered from either of the quadratic equations mentioned in corollary 1. The proper values x>f the co-efficients being written in the Srft/ofvthem, it becomes x z ~.l x — -=o, where one value of ~x is — ^— I =- 2 . The- 3 3 3 other quadratic becomes xf - " x -F 2 =0, one of whofe roots is 4. a •■ 8 EX A..M.P Iv E H. It being known that the equation x* - x* ~jx % 4- 1 3* - 6 = o has two equal roots, to find them. Here p-i, q= -7, r=. -1& and j-~6; and A = 59- 59 39 1 39 1 23009' " ^ 12800 j D— B 12800 r , - , 5io6i' and aT(J = TJboo= '» 0ne of the roots fou § ht - The 42 2 Mr. helli nS*s Method of finding the The fame value of x may be found from either of the two general quadratic equations given in corollary i. From the fir ft of them we get one -value, of x.= 21z — Lil = i , And from 6 59 the other, one value ~ iZ2Z_^I2ii which is alfo— i. ,39 « E 'X A M P L E III. Given the equation at 4 — I x + ~~ — o, in which two values 1 2 16 of .v are equal to each other, to find them. By corollary %, we .have x^= lip-^liii ..s= -. I . By corol. 2, jo 2 io a ^ J 3/1 1 T H E O R E M HI. i/ tbe fur/olid equation x y -px* + qx 3 - rx % + sx - 1 = o has two roots equal to each other, and you make A =*l¥^L q , B = *P r ~ 2Q! ' 4/>i>-l0?' 4pp~lOq* 4pp-:ioq> ' S& + 4P* 5 A + 4/>' 5A + 4/.' A-u» ^ ^ ATI) » * ^ A^G ' ^^ ^ ~ A^G ' ^ W -^^ 0iW ^ ^ ^?/d/ values of x bez= ~~ • ^ 1 — Cr The inveftigation of this theorem being altogether fimilar to that of the laft, it is unneceflary to give it here. The .difference of equations being taken as in the inveftiga- tion of theorem II. it will appear, that one of the equal roots may alfo be had from any one of the following five equations, of which fometimes one, fometimes another, will be the moft eligible J. equal Roots of an Equation by Divifion. 423 i . 5# 4 - 4^x 3 + $qx % ~ zrx + S — Q. 2, fix* — zqx 3 •{■yx 1 -4.SX + 5/ = o,- 3. a; 5 + Ax* + Bx + C = o. 4. . # 5 4- D,v* 4- E# - F = 0^ 5. x 2 + Gx + H = o. It is obvious, that, when fi vanifhes, the work will be 'con- siderably fhortened; and when both p and q are wanting, though the above formula fails, yet the equal root may beealily obtained from the equation fix" *-zqx % + 3 rx 1 - ^sx + st = o, which in, that cafe becomes - $rx x — 4SX 4- 5^ = 0. Whenever s is wanting, F, in the fecond cubic above, will be — o, and eonfequently x may be found from the quadratic equation xl 4- Da: 4- E = o. But in any of thefe cafes the equal root may be found by divifion. However, the operation probably will not, in general, be fo ihort as- extracting the root of the quadratic ; I will therefore haften to give an. example or two of the u.fe of the theorem. E'Xi(4'M PL E: I. Given x- +ar 5 — ** +o , 09433 — O, to find x, two values of it being equal to each other.; Hereto, q=i, r=i-i, j~q, ./=•=■ .-©'09433, and we get A = - 1 -5 F — o B — o G= -0*2231 C — +0*23.58 H~ -0*1241 D= +0*4 I '= -0-0972 £=-0*4238 K = -0*185 and x —■ t~zq — 0*48. The proper values of the co-efficients being written in the- fee equations before. mentioned* and fome of -them divided by the* 424 Mr. hellins's Method of finding the the co-efficient of the higheft power of x, we have thefe four equations, in each of which one' value of x is one of the equal ones fougirt : # 3 # + c6x — 0*4—0. x z - i'5x** + 0*2358 = 0. M* + -4.x — 0*4238 ==0. A; i -o , 2 23ix — 0-1241 =0. Now the moft eligible equation 4sthe quadratic -•#* 4-0-4* -— 0-4238 = o, whofe affirmative root is ^0-4638) - 0*2 = 0*48 1 1, agreeing with the value of x found above, but-true to two places lower hr the decimal. EX AM PL E II. To find the two equal -values of at in the equation 64A* 5 — 2o# 2 4- 3 =.0. The given equation -being divided by 64, we have jc 5 -0-3125^ + 0-046875 = ; and then, from the firft of the five equations given above, we get 5X 4 - 0*625^ = 0, and# = ^0-125 = 0*5. .-But from the fecond of the equations juft mentioned, we have 0*9375;*** - 0*234375 = o, or x z = 21M1375— 0-2 c,*and x— */o*2 5 = o- c. From the foregoing few >pages it is evident, that rules may- be made for finding the equal roots of equations of more than live dimensions by divifion ; but the operations by them will, in moft cafes, be, long and tedious. <It is obvious, however, that fuch equations may be depreffed to any dimenlion the alge- braift pleafes. JL It has indeed been-fuppofed, that the number of equations that have equal roots is but fmall, and, confequently, that the chief equal Roots of < an liquation by Dh if on. 425' chief ule of the rules for finding their roots is to get limits and approximations to the roots of equations in general. That ufe, it muft be allowed, were it the only one, is fufficie.nt to pay for inveftigating them. But if the equations that have equal roots mould hereafter be found not fo few as has been generally received, then the ufe of the above theorems will become more extenfive. I beg leave to add, that this mort effay is but a fmall part of a, work, in which,, if 1 mould ever have leifure to put a finish- ing hand to it, fcmething more on this fubject may very pro- bably appear. In the mean while, I hope, this little piece willi be candidly received by thofe who have more leifure and, better abilities; for ftudies of this kind. Cbnftantine, lebraary 9, 1782. Vol. LXXIi; K" K K.